In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.

Let X be a complex Banach space and denote by ${\cal L}(X)$ the Banach algebra of all bounded linear operators on X. We prove that if φ: ${\cal L}(X) \to {\cal L}(X)$ is a linear surjective map such that for each $T \in {\cal L}(X)$ and x ∈ X the local spectrum of φ(T) at x and the local spectrum of T at x are either both empty or have at least one common value, then φ(T) = T for all $T \in {\cal L}(X)$. If we suppose that φ always preserves the modulus of at least one element from the local spectrum, then there exists a unimodular complex constant c such that φ(T) = cT for all $T \in {\cal L}(X)$.

A ${{C}^{*}}$-algebra Ahas the ideal property if any ideal $I$ of $A$ is generated as a closed two-sided ideal by the projections inside the ideal. Suppose that the limit ${{C}^{*}}$-algebra $A$ of inductive limit of direct sums of matrix algebras over spaces with uniformly bounded dimension has the ideal property. In this paper we will prove that $A$ can be written as an inductive limit of certain very special subhomogeneous algebras, namely, direct sum of dimension-drop interval algebras and matrix algebras over 2-dimensional spaces with torsion ${{H}^{2}}$ groups.